The Dynamics of Growing Biofilm J. P. Keener (Utah), with lots of help from Nick Cogan (Tulane), Jack Dockery (Montana SU) and the NSF U U The Power of Mathematics The Excitement of Biology

Biofilms Biofilm fouling of fiber filter Placque on teeth U U The Power of Mathematics The Excitement of Biology

Some of the Interesting and Important Questions P. Aeruginosa (on catheters, IV tubes, etc.) How do gels grow? Mucus secretion (bronchial tubes, stomach lining) Colloidal suspensions of cells (cancer cell growth) Gel Morphology (The cellular shape of sponges) Why are gels important? Protective capability High viscosity (low washout rate) -for drugs, acid protection

Biofilm formation in Pseudomonas Aeruginosa Wild typeBiofilm mutantMutant with autoinducer Heterogeneous structures

The Dynamics of Growing Biofilm Quorum Sensing: –What is it? –How does it work? Heterogeneous structures: –How do these cells use polymer gel (EPS) for locomotion? –What are the mechanisms of pattern (structure) formation? –Why is polymer gel so effective as a protective environment?

Quorum Sensing in P. Aeruginosa P. Aeruginosa Major cause of hospital infection in the US. Major cause of deaths in intubated CF patients, and IV fed patients. Quorum Sensing: The ability of a bacterial colony to sense its size and regulate its activity in response. Examples: Vibrio fisheri, myxococcus, P. Aeruginasa P. Aeruginosa in planktonic (non-colonized) form are non-toxic, but as a biofilm, they are highly toxic and well protected by the polymer gel in which they reside. However, they do not become toxic or begin to form polymer gel until the colony is of sufficient size to overwhelm the immune system. Before this, they cannot be detected by the immune system.

Quorum sensing in P. Aeruginosa Planktonic Loosely BoundEPS secreting

“Wall Sensing” in P. Aeruginosa Wall Sensing: The ability of bacteria to differentiate in response to Contact with a wall (the substratum). PlanktonicLoosely Bound EPS secreting

The Dynamics of Growing Biofilm Quorum Sensing: –What is it? –How does it work? Heterogeneous structures: –How do these cells use polymer gel for locomotion? –What are the mechanisms of pattern (structure) formation? –Why is polymer gel so effective as a protective environment?

The Biochemistry of Quorum Sensing ALasR lasR lasI LasRLasI A 3-oxo-C12-HSL rsaL LasR B C4-HSL RhrRB RhlI rhlI GacA Vfr rhlR RhlR

An ODE Model VariableConcentration RLasR A3-oxo-C12-HSL PDimer complex LLasI SRsaL rlasR mRNA llasI mRNA srsaL mRNA lasI LasI ALasR lasR LasR A 3-oxo-C12-HSL rsaL LasR

Modeling Biochemical Reactions with ODE’s Rate of productionRate of degradation Rate of productionRate of degradation Production of enzyme from mRNA lasI LasI Bimolecular reaction ALasR A 3-oxo-C12-HSL Production of mRNA

The full system of differential equations for the biochemistry: lasI LasI ALasR lasR LasRA 3-oxo-C12-HSL rsaL LasR

Modeling diffusion across cell membrane Does this model exhibit quorum sensing? Two ways to proceed: 1) Numerical simulation (but few of the 22 parameters are known) 2) Qualitative analysis A E where is the cell volume fraction.

Quasi-steady state analysis: Assume that all “fast” variables are in quasi-steady state

An even simpler model Set R=A (which is not correct), to find

A Simple Model for Wall Sensing Cells are immobile, so internal variables A and V do not diffuse, but extracellular autoinducer E diffuses (rapidly) so L E x

A Simple Model for Wall Sensing - cont’d We can solve for E(x=L) assuming a quasi-equilibrium, to find Since p decreases as L 0, this has a chance of up-regulating for small L.

A Proper (PDE) Model Cells are immobile, so internal variables A and V do not diffuse, but extracellular autoinducer E diffuses so on the domain, subject to the (Robin) boundary condition on

Autoinducer with fixed colony size, variable density

Autoinducer concentration; fixed density, variable size colony

Heterogeneous Structures How do these cells use polymer gel for locomotion? What are the mechanisms of pattern (structure) formation? Why is polymer gel so effective as a protective environment?

A Hydrogel Primer What is a hydrogel? A tangled polymer network in solvent Examples of biological hydrogels: Micellar gels Jello (a collagen gel ~ 97% water) Extracellular matrix Blood clot Mucin - lining the stomach, bronchial tubes, intestines Gycocalyx - lining epithelial cells of blood vessels Sinus secretions Other examples of gel-like structures Cell colonies - colloidal suspensions Gel morphology - sponges, jellyfish

Fibrin Network (Blood Clot)

A Hydrogel Primer - II Function of a biological hydrogel Decreased permeability to large molecules Structural strength (for epithelial cell walls) Capture and clearance of foreign substances Decreased resistance to sliding/gliding High internal viscosity (low washout) Important features of gels Usually comprised of highly polyionic polymers Often exhibit large volumetric changes eg. Highly compressed in secretory vessicle and expand rapidly and dramatically on release Can undergo volumetric phase transitions in response to ionic concentrations (Ca++, H+), temperature,.. Volume is determined by combination of attractive and repulsive forces: -- repulsive electrostatic, hydrophobic -- attractive, hydrogen binding, cross-linking

How gels grow Polymerization/deposition (blood clots) Secretion High Ca++ Ca++Na+ Highly condensed Decondensed (swelled) Interesting facts: - Diffusion coefficient can vary substantially as a function of Ca++ - Expansion does not occur in distilled water.

Modeling Biofilm Growth A two phase material with polymer volume fraction Network Phase (EPS) Solute Phase Bacterial ConcentrationResource

Force Balance Solute phase (solute-network friction) (viscosity) (pressure) Network phase (pressure)(solute-network friction)(osmosis) Incompressibility Remark: Models of this form for the network phase have been used by lots of people (Dembo and He, Lubkin and Jackson, Wolgemuth, Oster, Mogilner, …) Remark 2: A better model might keep track of Ca++ and Na+ concentrations as well

Osmotic Pressure What is the meaning of the term In some formulations (osmosis) ? whereis the Free Energy > 0 gives expansion (swelling) < 0 gives contraction (deswelling) { From Flory-Huggins theory

Phase Separation Double-welled potentials give phase transitions and phase separation, similar to Cahn-Hilliard To maintain an edge,must be of the form (This is a theorem)

Movement by Swelling

Propulsion by Swelling

No gel; no swell

Heterogeneous Structures How do these cells use polymer gel for locomotion? What are the mechanisms of pattern (structure) formation? Why is polymer gel so effective as a protective environment?

“Limited Resource” Instability Remark: This instability does not occur in a resource rich environment.

Channeling A Pouiseille flow

Channeling

Conclusions Quorum Sensing: –What is it? A biochemical switch –How does it work? Extracellular autoinducer fails to diffuse away. Heterogeneous structures: –How do these cells use polymer gel for locomotion? Expansion by swelling –What are the mechanisms of pattern (structure) formation? Limited resource fingering Friction driven channeling Others?(Is sloughing related to a Kelvin-Helmholz instability?) –Why is EPS so effective as a protective environment? Restricted pore size, chemical reactivity???

More Questions (than answers) Why does P.Aeruginosa attack only immune-compromised individuals? (burn victims, post-surgery) Why does P.Aeruginosa find a ready host in CF patients? (Decreased diffusion???) What is it about the CFTR mutation in CF patients that makes their mucin so thick? Is there cross talk between P. Aeruginaosa and other pathogens and if so, how?